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Solve $\frac{x+\dots+x^K}{K} = \frac{1}{2}$ for large values of $K$

I am interested in the unique solution $x$ for the equation : $$ p_K(x)=\frac{x+\dots+x^K}{K}=\frac{1}{2}, $$ for large values of $K$. When $K$ is small ($K=1$ and $K=2$) we can solve this equation explicitly and find : $$ x=\frac{1}{2}, \frac{\sqrt{5}-1}{2}. $$ For $K=3$ we still get an explicit solution which is more complicated and from $K=4$ […]

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Ask Mathematics

Solve $\frac{x+\dots+x^K}{K} = \frac{1}{2}$ for large values of $K$

I am interested in the unique solution $x$ for the equation : $$ p_K(x)=\frac{x+\dots+x^K}{K}=\frac{1}{2}, $$ for large values of $K$. When $K$ is small ($K=1$ and $K=2$) we can solve this equation explicitly and find : $$ x=\frac{1}{2}, \frac{\sqrt{5}-1}{2}. $$ For $K=3$ we still get an explicit solution which is more complicated and from $K=4$ […]